Abstract
We consider the nonlinear problem arising in population dynamics: - + u(t) p = λu(t), u(t) > 0, t ∈ I:= (0,1), u(0) = u(1) = 0, where p > 1 is a constant and λ > 0 is a positive parameter. We establish the crucial asymptotic formula for the branch of positive solutions λ q (α) in L q -framework as α → oo, where a:= ∥u λ ∥ q (1 < q < ∞). Especially, for the original logistic equation, namely the case where p = 2 and q = 1, we obtain not only the asymptotic expansion formula for λ 1 (α) but also the remainder estimate. Such a formula for the bifurcation branch seems to be new.
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